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We consider a degenerate parabolic system which models
the evolution of nematic liquid crystal with variable degree of orientation.
The system
is a slight modification
to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of
Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120].
We prove the global existence
of weak solutions by passing to the limit in a regularized system.
Moreover, we
propose a practical...
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
A high-order compact finite difference scheme for a fully nonlinear
parabolic differential equation is analyzed. The equation arises in the
modeling of option prices in financial markets with transaction costs.
It is shown that the finite difference solution converges locally
uniformly to the unique viscosity solution of the continuous equation.
The proof is based on a careful study of the discretization matrices and on
an abstract convergence result due to Barles and Souganides.
We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.
We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...
We present here a discretization of
a nonlinear oblique
derivative boundary value problem for the heat equation in dimension
two.
This finite difference scheme takes advantages of the
structure of the boundary condition, which can be reinterpreted as a
Burgers equation in the space variables. This enables to obtain an
energy estimate and to prove the convergence of the scheme.
We also provide some numerical simulations of this
problem and a numerical study of the stability of the scheme, which
appears...
We study high-order numerical methods for solving Hamilton-Jacobi equations. Firstly, by introducing new clear concise nonlinear weights and improving their convex combination, we develop WENO schemes of Zhu and Qiu (2017). Secondly, we give an algorithm of constructing a convergent adaptive WENO scheme by applying the simple adaptive step on the proposed WENO scheme, which is based on the introduction of a new singularity indicator. Through detailed numerical experiments on extensive problems including...
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...
In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.
This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial...
We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition , we obtain error estimates in of order where is the degree of the local polynomials.
We present the convergence analysis of locally divergence-free discontinuous Galerkin methods
for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition
, we obtain error
estimates in L2 of order where m is the degree of the local polynomials.
The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.
We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not...
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